A measurement of the cross section for the combined two-photon production of charged pion and kaon pairs is performed using 1.2~$\rm fb^{-1}$ of data collected by the CLEO~II detector at the Cornell Electron Storage Ring. The cross section is measured at invariant masses of the two-photon system between 1.5 and 5.0 GeV/$c^2$, and at scattering angles more than $53^\circ$ away from the $\gamma\gamma$ collision axis in the $\gamma\gamma$ center-of-mass frame. The large background of leptonic events is suppressed by utilizing the CsI calorimeter in conjunction with the muon chamber system. The reported cross section is compared with leading order QCD models as well as previous experiments. In particular, although the functional dependence of the measured cross section disagrees with leading order QCD at small values of the two-photon invariant mass, the data show a transition to perturbative behavior at an invariant mass of approximately 2.5 GeV/$c^2$. hardcopies with figures can be obtained by writing to to: Pam Morehouse preprint secretary Newman Lab Cornell University Ithaca, NY 14853 or by sending mail to: preprints@lns62.lns.cornell.edu
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